Elliptic curve with LMFDB label 115920.ep1 (Cremona label 115920er1)

• Label: 115920.ep1
• Conductor: $115920$
• Discriminant: $2.744\times 10^{24}$
• j-invariant: \( \frac{2625564132023811051529}{918925030195200000} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-41386827x+64425512954\)

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(-697, 304850\right)\)
$\hat{h}(P)$	≈	$5.2813532302471566600084353313$

Torsion generators

\( \left(1669, 0\right) \)

Integral points

\((-697,\pm 304850)\), \( \left(1669, 0\right) \)

Invariants

Conductor:	\( 115920 \)	=	$2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$
Discriminant:	$2743895437362384076800000 $	=	$2^{42} \cdot 3^{11} \cdot 5^{5} \cdot 7^{2} \cdot 23 $
j-invariant:	\( \frac{2625564132023811051529}{918925030195200000} \)	=	$2^{-30} \cdot 3^{-5} \cdot 5^{-5} \cdot 7^{-2} \cdot 23^{-1} \cdot 1979^{3} \cdot 6971^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$3.3907035350110211489708101835\dots$
Stable Faltings height:	$2.1482502101170209938559554436\dots$

BSD invariants

Analytic rank:	$1$
Regulator:	$5.2813532302471566600084353313\dots$
Real period:	$0.074122273445691298079878501637\dots$
Tamagawa product:	$ 80 $  = $ 2^{2}\cdot2\cdot5\cdot2\cdot1 $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L'(E,1) $ ≈ $ 7.8293181659132956046825624237 $

Modular invariants

Modular form 115920.2.a.ep

\( q + q^{5} + q^{7} - 2 q^{13} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

Modular degree:	13824000
$ \Gamma_0(N) $-optimal:	yes
Manin constant:	1

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$2$	$4$	$I_{34}^{*}$	Additive	-1	4	42	30
$3$	$2$	$I_{5}^{*}$	Additive	-1	2	11	5
$5$	$5$	$I_{5}$	Split multiplicative	-1	1	5	5
$7$	$2$	$I_{2}$	Split multiplicative	-1	1	2	2
$23$	$1$	$I_{1}$	Non-split multiplicative	1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2B	2.3.0.1

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	add	split	split	ss	ord	ss	ss	nonsplit	ord	ord	ord	ss	ord	ord
$\lambda$-invariant(s)	-	-	2	6	1,1	1	1,3	1,1	1	1	1	1	1,1	1	1
$\mu$-invariant(s)	-	-	0	0	0,0	0	0,0	0,0	0	0	0	0	0,0	0	0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 115920.ep consists of 2 curves linked by isogenies of degree 2.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{345}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.0.4327680.7	\(\Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.